Average Error: 7.9 → 5.7
Time: 9.6s
Precision: 64
\[x0 = 1.854999999999999982236431605997495353222 \land x1 = 2.090000000000000115064208161541614572343 \cdot 10^{-4} \lor x0 = 2.984999999999999875655021241982467472553 \land x1 = 0.01859999999999999847899445626353553961962\]
\[\frac{x0}{1 - x1} - x0\]
↓
\[\frac{\frac{x0}{1 - x1} \cdot \frac{\sqrt{x0}}{\frac{1 - x1}{\sqrt{x0}}} - x0 \cdot x0}{\left(\sqrt[3]{x0 + \frac{x0}{1 - x1}} \cdot \sqrt[3]{x0 + \frac{x0}{1 - x1}}\right) \cdot \sqrt[3]{x0 + \frac{x0}{1 - x1}}}\]
\frac{x0}{1 - x1} - x0
↓
\frac{\frac{x0}{1 - x1} \cdot \frac{\sqrt{x0}}{\frac{1 - x1}{\sqrt{x0}}} - x0 \cdot x0}{\left(\sqrt[3]{x0 + \frac{x0}{1 - x1}} \cdot \sqrt[3]{x0 + \frac{x0}{1 - x1}}\right) \cdot \sqrt[3]{x0 + \frac{x0}{1 - x1}}}
double f(double x0, double x1) {
        double r155201 = x0;
        double r155202 = 1.0;
        double r155203 = x1;
        double r155204 = r155202 - r155203;
        double r155205 = r155201 / r155204;
        double r155206 = r155205 - r155201;
        return r155206;
}
↓
double f(double x0, double x1) {
        double r155207 = x0;
        double r155208 = 1.0;
        double r155209 = x1;
        double r155210 = r155208 - r155209;
        double r155211 = r155207 / r155210;
        double r155212 = sqrt(r155207);
        double r155213 = r155210 / r155212;
        double r155214 = r155212 / r155213;
        double r155215 = r155211 * r155214;
        double r155216 = r155207 * r155207;
        double r155217 = r155215 - r155216;
        double r155218 = r155207 + r155211;
        double r155219 = cbrt(r155218);
        double r155220 = r155219 * r155219;
        double r155221 = r155220 * r155219;
        double r155222 = r155217 / r155221;
        return r155222;
}

Error

Bits error versus x0

Bits error versus x1

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.9
Target0.2
Herbie5.7
\[\frac{x0 \cdot x1}{1 - x1}\]

Derivation

  1. Initial program 7.9

    \[\frac{x0}{1 - x1} - x0\]
  2. Using strategy rm

  3. Applied flip--7.3

    \[\leadsto \color{blue}{\frac{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}{\frac{x0}{1 - x1} + x0}}\]

  4. Simplified7.3

    \[\leadsto \frac{\frac{x0}{1 - x1} \cdot \frac{x0}{1 - x1} - x0 \cdot x0}{\color{blue}{x0 + \frac{x0}{1 - x1}}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt7.3

    \[\leadsto \frac{\frac{x0}{1 - x1} \cdot \frac{\color{blue}{\sqrt{x0} \cdot \sqrt{x0}}}{1 - x1} - x0 \cdot x0}{x0 + \frac{x0}{1 - x1}}\]

  7. Applied associate-/l*5.6

    \[\leadsto \frac{\frac{x0}{1 - x1} \cdot \color{blue}{\frac{\sqrt{x0}}{\frac{1 - x1}{\sqrt{x0}}}} - x0 \cdot x0}{x0 + \frac{x0}{1 - x1}}\]
  8. Using strategy rm

  9. Applied add-cube-cbrt5.7

    \[\leadsto \frac{\frac{x0}{1 - x1} \cdot \frac{\sqrt{x0}}{\frac{1 - x1}{\sqrt{x0}}} - x0 \cdot x0}{\color{blue}{\left(\sqrt[3]{x0 + \frac{x0}{1 - x1}} \cdot \sqrt[3]{x0 + \frac{x0}{1 - x1}}\right) \cdot \sqrt[3]{x0 + \frac{x0}{1 - x1}}}}\]

  10. Final simplification5.7

    \[\leadsto \frac{\frac{x0}{1 - x1} \cdot \frac{\sqrt{x0}}{\frac{1 - x1}{\sqrt{x0}}} - x0 \cdot x0}{\left(\sqrt[3]{x0 + \frac{x0}{1 - x1}} \cdot \sqrt[3]{x0 + \frac{x0}{1 - x1}}\right) \cdot \sqrt[3]{x0 + \frac{x0}{1 - x1}}}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (x0 x1)
  :name "(- (/ x0 (- 1 x1)) x0)"
  :precision binary64
  :pre (or (and (== x0 1.855) (== x1 2.09000000000000012e-4)) (and (== x0 2.98499999999999988) (== x1 0.018599999999999998)))

  :herbie-target
  (/ (* x0 x1) (- 1 x1))

  (- (/ x0 (- 1 x1)) x0))